Hop Domination in Graphs-II

Let G = (V;E) be a graph. A set S ⊂ V (G) is a hop dominating set of G if for every v ∈ V - S, there exists u ∈ S such that d(u; v) = 2. The minimum cardinality of a hop dominating set of G is called a hop domination number of G and is denoted by γ_h(G). In this paper we characterize the family of trees and unicyclic graphs for which γ_h(G) = γ_t(G) and γ_h(G) = γ_c(G) where γ_t(G) and γ_c(G) are the total domination and connected domination numbers of G respectively. We then present the strong equality of hop domination and hop independent domination numbers for trees. Hop domination numbers of shadow graph and mycielskian graph of graph are also discussed.